Annus MirabilisEinstein in 1905
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light quanta

The equation explained

RN8πL3=αβ\frac{R}{N} \frac{8 \pi}{L^3} = \frac{\alpha}{\beta}
In wordsThe gas constant over the number of molecules in a gram-equivalent, times eight π over the speed of light cubed, equals the constant α over the constant β.OverviewSection 1's density and the long-wave limit of Planck's formula have the same form, so their coefficients must be equal. This joins the gas theory's R/N to the radiation constants α and β.Full explanationBoth expressions say the density grows as ν2T\nu^2 T; they differ only in the number in front. Section 1 wrote it as RN8πL3\frac{R}{N} \frac{8 \pi}{L^3}, from Maxwell's theory, the electron theory and the kinetic theory of gases; Planck's formula at long waves writes it as αβ\frac{\alpha}{\beta}, from measurement. The paper sets the two equal. The step leans only on the region where section 1's picture agrees with experience, long waves and high densities, and not on Planck's theory of black radiation: this is the independence, 'to a certain degree', that section 2 set out to show.Every stepϱν=RN8πν2L3T\varrho_\nu = \frac{R}{N} \frac{8 \pi \nu^2}{L^3} TSection 1's density (its seventh display).ϱν=αβν2T\varrho_\nu = \frac{\alpha}{\beta} \nu^2 TPlanck's formula at long waves (the preceding display).RN8πL3ν2T=αβν2T\frac{R}{N} \frac{8 \pi}{L^3} \nu^2 T = \frac{\alpha}{\beta} \nu^2 TBoth describe the same density where both hold, so set them equal, writing section 1's 8πν28 \pi \nu^2 as 8π8 \pi times ν2\nu^2.RN8πL3=αβ\frac{R}{N} \frac{8 \pi}{L^3} = \frac{\alpha}{\beta}Divide both sides by ν2T\nu^2 T, the same on both: the coefficients must agree. The printed line.Also

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